# .

**Griechische Wissenschaftler**

(a short version of a biography written by Stavros G. Papastavridis )

CHRISTOS PAPAKYRIAKOPOYLOS (Χρήστος Παπακυριακόπουλος) was born in ATHENS(-Chalandri) in 1914. His father, coming from PELOPONESE (TRIPOLIS), was an affluent merchant. He graduated from Varvakeion, (*the most prestigious, at that time and for many decades later, high school in Greece*). He enrolled in the National Metsovion Institute of Technology (Ethniko Metsovion Polytexneio) in 1933. There he met Professor of Mathematics Nikolaos Kritikos, who influenced him to switch and enroll in the Mathematics Department of the University of Athens.

His interest was attracted to Algebraic Topology, (subject having minimal activity in Greece at that time), and he read on his own Aleksandrov and Hopf, Topologie, which has appeared in 1935. *(Pavel Sergeevich Aleksandrov, Born: 7 May 1896 in Bogorodsk (also called Noginsk), Russia, Died: 16 Nov 1982* *in Moscow, USSR. Heinz Hopf, Born: 19 Nov 1894* *in Breslau, Germany* *(now Wroclaw, Poland) Died: 3 June 1971* *in Zollikon, Switzerland. **From 1926 Aleksandrov and Hopf were close friends working together. They spent some time in 1926 in the south of France* *with Neugebauer. Then Aleksandrov and Hopf spent the academic year 1927-28 at Princeton* *in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton* *and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes).*

His first paper was

Papakyriakopulos, Ch.: Ueber eine Indicatrix der ebenen geschlossenen Jordankurven.(Greek) Bull. Soc. Math. Greece 18(1938) 84-92.

He got his Doctor's degree from the Mathematics Department of the University of Athens at 1943, after recommendation from Constantin Caratheodory. In his thesis he gave a new proof of the topological invariance the homology groups for Simplicial Complexes. His thesis was published

Papakyriakopoulos, Ch.: Ein neuer Beweis fuer die Invarianz der Homologiegruppe eines Komplexes. (Greek) Bull. Soc. Math. Greece 22(1946) 1-154.

*(He provided another proof of the so called Hauptvermutung (Main assumption), which was proved for the first time by James Waddell Alexander, (Born: 19 Sept 1888* *in Sea Bright, New Jersey, USA, Died: 23 Sept 1971* *in Princeton, New Jersey, USA. In collaboration with **Veblen, he showed that the topology of manifolds could be extended to polyhedra. Before 1920 he had shown that the homology of a simplicial complex is a topological invariant. Alexander's work around this time went a long way to put the intuitive ideas of Poincaré on a more rigorous foundation).*

During those years he was working as an (non-paid) Teaching Assistant of Professor N. Kritikos, at the National Metsovion Institute of Technology (Ethniko Metsovion Polytexneio).

In the national referendum of 1935, he voted openly against the return of the king. In the harsh years of the Nazi occupation he joint the National Front of Liberation. After the civil war clashes of December 1944 in Athens, he followed the forces of National Front of Liberation in the countryside, where he found himself in Karditsa teaching Elementary arithmetic to Primary school students. At this time his godfather was minister of Interior, looking for Papakyriakopoulos in order to appoint him Mayor of XALANDRI, flabbergasted to find out that Papakyriakopoulos was in the mountains with the guerillas! At about the same time his brother died in action fighting with the loyal to the Greek government in exile forces, the so called Rimini brigade, as part of the allied push in northern Italy. It sounds like an ancient Greek tragedy, almost like Eteoklis and Polynikis, In ANTIGONE or SEVEN AGAINST THEBES. The terrible political division that brought to Greece an extremely destructive civil war 1946-49, with deep repercussions in the following decades, has cut across the Papakyriakopoulos family. The Varkiza agreement in February 1944, among the fighting political factions, gave a temporary breath to peace in Greece and Papakyriakopoulos returned to Athens and at the National Metsovion Institute of Technology (Ethniko Metsovion Polytexneio). There the political climate was unfavorable to National Front of Liberation sympathizers, eventually Professor Kritikos was fired (to be rehired later on in the 50ies, when things have calmed down a little bit), and Papakyriakopoulos was forced to leave in 1946.

Working on his own with no connection to any mathematical center outside Greece, he concentrated on the low dimensional topology, and among other things Dehn's lemma caught his fancy. Max Wilhelm Dehn, (Born: 13 Nov 1878 in Hamburg, Germany, Died: 27 June 1952 in Black Mountain, North Carolina, USA), was a student of Hilbert and a solver of Hilbert**'**s 3rd problem, (from the famous least of 23 problems). *( Hilbert 's 3rd problem, is the following. The equality of two volumes of two tetrahedra of equal bases and equal altitudes. In two letters to Gerling, Gauss expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid*

*just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra).*In 1910, in M.Dehn, "Uber die topologie des dreidimendionalen Raumes, Math. Ann. 69(19100, pp. 137-168, Dehn gave a proof of a lemma concerning loops in three dimensional manifolds. In 1929 H. Knesser realized a gap in Dehn's proof. Papakyriakopoulos tried to close the gap, and he sends his purported proof to a very distinguished Princeton Knot Theorist, Ralph Fox. Fox found a gap in Papakyriakopoulos proof, but he was very favorably impressed by the young Papakyriakopoulos who was working in total scientific isolation, and he urged him to come to Princeton. Papakyriakopoulos was always recognizing the importance of this encouragement and subsequent support that he received from R. Fox. Papakyriakopoulos went to Princeton at 1948, never to return to Greece again, except for a very short visit in 1952 to attend the funeral of his father.

The Greek Security Police pursue him in the USA, trying to convince the USA Immigration authorities to expel him from the country. Princeton University supported him, and Papakyriakopoulos was always very grateful for that. (*We may note in passing that the list of people who found political asylum at Princeton University, includes Albert Einstein and Thomas Mann in the 30ties and *Chai Ling *(student leader of the Tiananmen Square uprising in 1989) in the 90ties*

FUNDAMENTAL NOTIONS

Below we provide some necessary terminology.

Let D^{n} = { (x_{1}, x_{2}, …, x_{n} ) Î R^{n} : x_{1}^{2 }+ x_{2}^{2} + …+ x_{n}^{2} £1 }be the closed n-disk, and its subsets S^{n} = { (x_{1}, x_{2}, …, x_{n} ) Î R^{n} : x_{1}^{2 }+ x_{2}^{2} + …+ x_{n}^{2} =1 }is the n-sphere, and O^{n} = { (x_{1}, x_{2}, …, x_{n} ) Î R^{n} : x_{1}^{2 }+ x_{2}^{2} + …+ x_{n}^{2} £1 }be the open n-disk(open ball).

DIFFERENTIABLE MANIFOLD

A topological space M is a C^{¥} n dimensional manifold, if it is endowed with a collection (U_{i} , f_{i} )_{iÎI} , where I is a non empty index set

a) the U_{i} 's constitute an open covering of M

b) the f_{i} 's are homeomorphisms f_{i} : O^{n} ® U_{i}

c) If i, j Î I and U_{i} Ç U_{j} is non empty, then the homeomorphism

f_{j}^{-1}×f_{i} : f_{i}^{-1}(U_{i} Ç U_{j}) ® f_{j}^{-1}(U_{i} Ç U_{j}) is C^{¥}^{ }differentiable, (in the usual sense of Calculus of several real variables)

A ( C^{¥} )^{ }n dimensional manifold will be called here simply Manifold.

Manifolds do appear in a variety of ways, e.g. the state space of mechanical systems, the set of solutions of systems of equations (minus "few" points) etc.

As Marston Morse put it,

"Any problem which is non-linear in character, which involves more than one coordinate system or more than one variable, or where structure is initially defined in the large, is likely to require considerations of topology and group theory for its solution. In the solution of such a problems classical analysis will frequently appear as an instrument in the small, integrated over the whole problem with the aid of topology or group theory", (The Calculus of Variations in the Large, AMS Coll. Publ. Vol. 18, New York 1934).

DIFFERENTIABLE FUNCTION

If M, N are manifolds then the continuous function f : M®N is called Differentiable if f "translated locally in terms of coordinates becomes C^{¥}^{ }differentiable, (in the usual sense of Calculus of several real variables)", i.e. more precisely : Let (U_{i} , f_{i} )_{iÎI} and (V_{i} , g_{i} )_{iÎJ} be the associated open coverings of M and N respectively . Then for any x in M, there is an open neighborhood U of x, an open neighborhood V of f(x), mÎ I, nÎ J, so that UÍU_{m} , VÍV_{n} , F(U)ÍV, such that the function

g_{n}^{-1}×f×f_{m} : f_{m}^{-1}(U)®g_{n}^{-1}(V), is C^{¥}^{ }differentiable, (in the usual sense of Calculus of several real variables).

DIFFEOMORPHISM

If M, N are differentiable manifolds then the continuous function f : M®N is called DIFFEOMORPHISM if it is differentiable and if there exists differentiable function

g : N®M so that gf=I_{M }and fg=I_{N} .

The central problem

CLASSIFICATION PROBLEM for COMPACT MANIFOLDS

A) Find necessary and sufficient conditions for two given compact manifolds to be HOMEOMORPHIC or DIFFEOMORPHIC

B) Describe all types of compact manifolds up to HOMEOMORPHISM or DIFFEOMORPHISM

The only compact 1 dimensional manifold, up to homeomorphism or diffeomorphism is S^{1}

HOMOTOPY, HOMOLOGY etc

Let I=[0, 1]

Let X, Y be topological spaces and f, g: X®Y be continuous maps.

We call f and g homotopic if there is a map F: X´I®Y, such that for all xÎX we have

F(x, 0)=f(x) and F(x, 1)=g(x)

HOMOTOPY EQUIVALENCE

Two topological spaces are called homotopy equivalent if there are maps f:X®Y and g:Y®X, such that gf is homotopic to I_{X} and fg is homotopic to I_{Y}

If X is a topological space then seqyences of groups are defined the

HOMOLOGY groups

H_{0}(X), H_{1}(X), H_{2}(X), H_{3}(X), … H_{n}(X),

the COHOMOLOGY groups

H^{0}(X), H^{1}(X), H^{2}(X), …H^{n}(X),

and the HOMOTOPY groups

π_{0}(Χ), π_{1}(Χ), π_{2}(Χ), …π_{n}(Χ),

Group π_{1}(Χ) is called Fundamental Group and was introduced by POINCARE(1904).

If X is a connected topological space, we select arbitrarily one point xÎX. In a way the elements of the Fundamental group can be presented as maps f: S^{1}®X , with f(1)=x.

The map f represents the zero element in π_{1}(Χ), if it can be extended as a continuous map F:D^{1}®X

THE THREE BIG THEOREMS

THE LOOP THEOREM

Papakyriakopoulos, C.D.: On solid tori. Proc. London Math. Soc., III. Ser. 7(1957) 281-299.

Let M be a three dimensional manifold with non-void boundary ¶M. Suppose that there is some closed loop f: S^{1}®¶M (with possible self intersection), which is homotopically zero in M, but NOT homotopically zero in ¶M. Then there exist a simple closed loop F: S^{1}®¶M sharing this property.

The proof was given under some orient ability assumption, which were later removed in

STALLINGS JOHN. On the loop theorem, Ann. of Math. 72(1960), pp. 12-19.

Almost immediately after the loop theorem, Papakyriakopoulos proved his natural companion known as Dehn's Lemma.

DEHN's LEMMA

Papakyriakopoulos, C.D.: On Dehn's lemma and the asphericity of knots. Ann. of Math., II. Ser. 66(1957) 1-26 .

Let M be a three dimensional manifold with non-void boundary ¶M. Suppose that there is some simple closed loop f: S^{1}®¶M (without self intersection), which is homotopically zero in M. Then there is an embedding F: D^{2}®M which extends f.

The "adventures" of the statement inspired some poetically inclined. The following limerick*, (Limerick* *must have five lines with aabba rhyme scheme. The beat must be anapestic (weak, weak, strong) with three feet in lines 1, 2, and 5 and 2 feet in lines 3 and 4)*, in J.Milnor's 60th birthday's conference, Michael Spivak attributed it to J.Milnor himself.

The perfidious lemma of Dehn was every topologist's bane 'til Christos Papakyriakopoulos proved it without any strain.

The SPHERE THEOREM

was proved in the same paper in the same paper with Dehn's Lemma. Papakyriakopoulos proof contained some extra assumptions, which were later removed by a slight modification of Papakyriakopoulos proof, in

WHITEHEAD J.H.C. On the sphere in 3-manifolds, Bull. AMS 64(1958), pp. 161-166.

Let M be a (closed) orientable 3 dimensional manifold, such that π_{2}M is not the trivial group. Then there is an imbedding S^{2}®M which is homotopically non zero.

He presented those results in the International Congress of Mathematics in 1958 in Amsterdam, in the invited address

Papakyriakopoulos, C.D.: Some problems on 3-dimensional manifolds. Bull. Am. Math. Soc. 64(1958) 317-335.

For the three theorems above, the AMERICAN MATHEMATICAL SOCIETY, in 1964 awarded him the VEBLEN prize in geometry.

*(This prize was established in 1961 in memory of Professor Oswald Veblen(Born: 24 June 1880* *in Decorah, Iowa, USA, Died: 10 Aug 1960* *in Brooklyn, Maine, USA) through a fund contributed by former students and colleagues. The fund was later doubled by the widow of Professor Veblen, bringing the fund to $2,000. The first two awards of the prize were made in 1964 and the next in 1966; thereafter, an award will ordinarily be made every five years for research in geometry or topology under conditions similar to those for the Bocher Prize. *

*The first award was given 1964 to C. D. Papakyriakopoulos for his papers, On Solid Tori, Annals of Mathematics, Series 2, volume 66 (1957), pp. 1-26, and On Dehn's lemma and the asphericity of knots, Proceedings of the National Academy of Sciences, volume 43 (1957), pp. 169-172. *

POINCARE's CONJECTURE

From the early 60ies till his death Papakyriakopoulos devoted his efforts to the POINCARE CONJECTURE

Consider a compact 3 dimensional manifold V. Is it possible for the fundamental group of V to be trivial, even though V is not homeomorphic to S^{3} ?

Answering the above question negatively, became to be known as the POINCARE CONJECTURE, which has inspired topologist ever since, leading to many false proof s by established mathematicians, (e.g. J. H. C. Whitehead in 1934). The Danish Mathematician Piet Hein warns: A problem worthy of attack proves its worth by hitting back. But the conjecture by focusing interest in the topology of manifolds contributed to many big advances in our understanding manifolds, with deep repercussions to other fields in mathematics, and quite often lead to surprising consequences. If someone looks at the list of Fields medalists, there are not many those about whom you can say that theories on manifolds have not play a very important role on his research, (and those are mostly the recipients of the first awards in 1936 and 1950)

Andrew J. Wiles (Special Tribute)

In the years of Poincare and probably till the 40ies, the Poincare Conjecture looked like the simplest open question concerning the classification of manifolds. Common sense would have dictate that there is no hope in classifying higher dimensional manifolds unless we classify first the 3 dimensional one, and that there is no hope in classifying 3 dimensional manifolds unless we settle either way the Poincare Conjecture.

Finally after 100 years the Russian mathematician Grigori Perelman of the Steklov Institute of Mathematics in St. Petersburg provided a proof that is not yet official but experts consider that it is most likely correct.

So Papakyriakopoulos concentrated his efforts almost exclusively to the Poincare Conjecture. He was fully aware that perhaps it was a "winner takes all situation" and calmly was accepting this risk, with sincere humility. **His life in Princeton** **can be described in one word: Spartan-Laconic-Doric.** He was very disciplined and very organized: 8.00 hours at the Student Center cafeteria for breakfast, 8.30 at his office, 11.30 Lunch, 12.30 in his office, 15.00 Tea in the common room of the department, reading of the New York Times. 4.00 Seminar or back to his office. His life was an example of search for the truth and the price associated with it.

The life o Papakyriakopoulos influences the literature.

*Uncle Petros and Goldbach's Conjecture*, by the Greek author Apostolos Doxiadis, a book that has been translated in almost 20 languages.

After the downfall of military junta in July 1975, he renewed his passport and he was contemplating a trip back to Greece. But fate has decided otherwise, and on June 29 1976, after been hospitalized for stomach cancer, Christos Papakyriakopoulos took the road, which has no end and no return.

PAPERS by CH. PAPAKYRIAKOPOULOS

1. Papakyriakopulos, Ch.: Ueber eine Indicatrix der ebenen geschlossenen Jordankurven.(Greek) Bull. Soc. Math. Greece 18(1938) 84-92.

2. Papakyriakopoulos, Chr.: Ueber die geschlossenen Jordanschen Kurven im $\bbfR_n$.(Greek) Bull. Soc. Math. Greece 19(1939) 97-126.

3. Papakyriakopoulos, Ch.: Ein neuer Beweis für die Invarianz der Homologiegruppe eines Komplexes. (Greek) Bull. Soc. Math. Greece 22(1946) 1-154.

4. Papakyriakopoulos, C.D.: On the ends of knot groups. Ann. of Math., II. Ser. 62(1955) 293-299.

5. Papakyriakopoulos, C.D.: On solid tori. Proc. London Math. Soc., III. Ser. 7(1957) 281-299.

6. Papakyriakopoulos, C.D.: On Dehn's lemma and the asphericity of knots. Ann. of Math., II. Ser. 66(1957) 1-26 .

7. Papakyriakopoulos C.D.: On Dehn's lemma and the asphericity of knots. Proc. Nat. Acad. Sc., Vol. 43(1957) 169-172.

8. Papakyriakopoulos, C.D.: On the ends of the fundamental groups of 3-manifolds with boundary. Commentarii Math. Helvet. 32(1957) 85-92.

9. Papakyriakopoulos, C.D.: Some problems on 3-dimensional manifolds. Bull. Am. Math. Soc. 64(1958) 317-335.

10. Papakyriakopoulos, C.D.: The theory of three-dimensional manifolds since 1950. Proc. Int. Congr. Math. 1958, 433-440 (1960).

11. Papakyriakopoulos, C.D.: A reduction of the Poincare conjecture to other conjectures Bull. Am. Math. Soc. 68(1962) 360-366.

12. Papakyriakopoulos, C.D.: A reduction of the Poincare conjecture to other conjectures. II Bull. Am. Math. Soc. 69(1963) 399-401.

13. Papakyriakopoulos, C.: A reduction of the Poincare conjecture to group theoretic conjectures. Ann. Math., II. Ser. 77(1963) 250-305.

14. Papakyriakopoulos, C.: Attaching 2-dimensional cells to a complex. Ann. Math., II. Ser. 78(1963) 205-222.

15. Papakyriakopoulos, C.D.: Planar regular coverings of orientable closed surfaces. Knots, Groups, 3-Manif.; Pap. dedic. Mem. R. H. Fox, 1975, 261-292.

**REFERENCES**

ATIYAH MICHAEL. The Geometry and Physics of knots. Camb. Univ. Press 1990.

ANDERSON T. MICHAEL. Scalar Curvature and Geometrization Conjectures for 3-manifolds. Comparison Geometry, MSRI Publications, Volume 30, 1997, pp. 49-82.

DEHN MAX. Uber die topologie des dreidimendionales Raumes, Math. Ann. 69(19100, pp. 137-168.

DONALSON S.K. An application of gauge theories to 4 dimensional topology. J. Diff. Geom. 18(1983), pp. 279-315.

EHRENFEST P. In what way does it become manifest in fundamental laws of Physics that space has three dimensions ? Proc. Amsterdam Acad. 20 (1917).

FREED D.S. and UHLENBECK K.K. Instantons and 4 dimensional manifolds. 2nd edition, Springer Verlag , New York 1991.

FREEDMAN MICHAEL. The topology of 4 dimensional manifolds. J. Diff. Geom. 17 (1982) pp. 357-453.

GABAI DAVID. Valentin's Poenaru program for the Poincare conjecture. Geometry, topology and Physics, pp. 139-166, Conf. Proc. in honour of Raoul Bott, ed. S.T.Yau, Lecture Notes Geom. Topology VI, Internat. Press Cambridge MA 1995.

GOMPF R. Three exotic R^{4}'s and other anomalies(!). J. Diff. Geom. 18(1983) pp. 317-328.

GOMPF R. An infinite set of exotic R^{4}'s . J. Diff. Geom. 18(1985) pp. 283-300.

POENARU VALENTINE. The three big Theorems of Papakyriakopoulos, Bulletin of Greek Math. Society 18(19770 pp. 1-7.

POENARU VALENTINE. A program for the Poincare conjecture and some of its ramifications. TOPICS IN LOW-DIMENSIONAL TOPOLOGY, In Honor of Steve Armentrout, Proceedings of the Conference on Low-Dimensional Topology *University Park, Pennsylvania, USA May 1996*, edited by A Banyaga, H Movahedi-Lankarani & R Wells , World Scientific.

SMALE S. Generalized Poincare Conjecture in dimensions greater than 4, Ann. of Math. 64(19600, pp. 399-405.

SMALE STEPHEN. The story of higher dimensional Poincare conjecture (what actually happened on the beaches of Rio), Math. Intel. bf 12 (19900 44-51.

STALLINGS JOHN. On the loop theorem, Ann. of Math. 72(1960), pp. 12-19.

STALLINGS JOHN. Polyhedral homotopy-spheres, BAMS, 66(1960), pp. 485-488.

TAUBES C.H. Gauge Theory on asymptotically periodic 4 manifolds. J. Diff. Geom. 25 (1987) pp. 363-430.

THURSTON W.P. Three dimensional Manifolds, Kleinian groups and hyperbolic geometry. The Mathematical Heritage of Henri Poincare. Proc. Symp. Pure Math. 39(1983), Part 1. (Also in Bull. Amer. math. Soc. 6(1982) 357-381.

THURSTON WILLIAM P. Three dimensional Geometry and Topology. Ed. Silvio Levy, Vol. 1, Princeton Math series 35, Princeton Univ. Press 1997.

THURSTON W.P. and Weeks Jeffrey R. The Mathematics of 3 dimensional Manifolds. Scient. Amer. July 1984, 251(1) pp. 94-106.

WHITEHEAD J.H.C. On the 2-sphere in 3-manifolds, Bull. AMS 64(1958), pp. 161-166.

ZEEMAN E.C. The Poincare Conjecture for n³5, TOPOLOGY of 3-MANIFOLDS (1961) pp. 198-204 Prentice Hall.

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